---
title: "Fourier transform property: Duality"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:ft-property-duality"
source: https://www.bctengineer.com/q/semester_6/communication_systems/ft-property-duality
site: BCT Engineering
---

# Fourier transform property: Duality

Communication Systems · Chapter 1: Introduction

- Asked in: 2067 Mangsir (CS I), 2 marks
- Video: [Duality Property of Fourier Transform](https://www.youtube.com/watch?v=9OK_i-n8gN8) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/ft-property-duality
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

State and explain the duality (symmetry) property of Fourier Transform.

## Short answer

Duality property: if g(t)↔G(f), then G(t)↔g(-f). Forward and inverse Fourier integrals differ only in exponent sign; swapping time and frequency roles yields duality. Example: rect(t/T)↔Tsinc(fT) gives Tsinc(tT)↔rect(f/T). Property halves required transform pairs.

## Answer

**Statement:** if $g(t)\leftrightarrow G(f)$, then the function $G$ used as a time signal has the transform

$$
G(t)\leftrightarrow g(-f)
$$

**Explanation:** the forward and inverse Fourier integrals differ only in the sign of the exponent. Swapping the roles of $t$ and $f$ in the inverse transform $g(t)=\int G(f)e^{j2\pi ft}df$ and replacing $t$ by $-f$ gives $g(-f)=\int G(t)e^{-j2\pi ft}dt$. For even $g(t)$, $G(t)\leftrightarrow g(f)$.

**Example:** since $\text{rect}(t/T)\leftrightarrow T\,\text{sinc}(fT)$, duality gives $T\,\text{sinc}(tT)\leftrightarrow\text{rect}(f/T)$; a sinc pulse in time has a rectangular (ideally band-limited) spectrum. Likewise $\delta(t)\leftrightarrow1$ gives $1\leftrightarrow\delta(f)$. The property halves the number of transform pairs one must remember.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. If g(t) <-> G(f), then G(t) <-> ?
   A. g(f)
   B. g(-f)
   C. -g(f)
   D. G(-f)
   Answer: B. Using G as a time function gives the transform g(-f); for even g(t) this is g(f). (Key point: Duality statement)

2. Duality holds because
   A. The forward and inverse Fourier integrals differ only in the sign of the exponent
   B. Fourier transforms are always real
   C. Signals are always even
   D. Time and frequency have the same units
   Answer: A. Swapping t and f in the inverse transform and reversing the sign produces the forward form. (Key point: Why duality holds)

3. Given rect(t/T) <-> T sinc(fT), duality gives that T sinc(tT) has the transform
   A. rect(f/T)
   B. T rect(f)
   C. sinc(f/T)
   D. delta(f)
   Answer: A. A sinc pulse in time has a rectangular, ideally band-limited spectrum. (Key point: Rect-sinc example)

## Related in Communication Systems

- [Fourier transform property: Modulation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-modulation) — Communication Systems, chapter 1
- [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1
- [Fourier transform property: Scaling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-scaling) — Communication Systems, chapter 1
- [Fourier transform property: Convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-convolution) — Communication Systems, chapter 1
- [Autocorrelation and ESD as Fourier transform pair](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aautocorrelation-esd-fourier-pair) — Communication Systems, chapter 1
- [Fourier transform properties; Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afourier-properties-rayleigh) — Communication Systems, chapter 1

## Related in other subjects

- [Duality property of CTFT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-duality) — Signals and Systems, chapter 3, asked 3×
- [Symmetry property of a circle](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acircle-symmetry-property) — Computer Graphics, chapter 2, asked 4×
- [FT of rectangular pulse and constant; duality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-rect-pulse-constant-duality) — Signals and Systems, chapter 3
- [Derive convolution property of z-transform](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-convolution-property-derive) — Digital Signal Analysis and Processing, chapter 2